A005996 G.f.: 2*(1-x^3)/((1-x)^5*(1+x)^2). (Formerly M1609)

2, 6, 16, 30, 54, 84, 128, 180, 250, 330, 432, 546, 686, 840, 1024, 1224, 1458, 1710, 2000, 2310, 2662, 3036, 3456, 3900, 4394, 4914, 5488, 6090, 6750, 7440, 8192, 8976, 9826, 10710, 11664, 12654, 13718

Offset

1,1

Comments

a(n) is also the number of triples (w,x,y) having all terms in {0,...,n} and w<R<=x, where R=max(w,x,y)-min(w,x,y). [Clark Kimberling, Jun 10 2012]

a(n) is also the sum of all elements of the square matrix M(n-1) = M1(n-1) x M2(n-1), where M1(n) is the square matrix with elements m1(i,j)= (1+(-1)^(i+j+1))/2, A057212; and M2(n) is the square matrix given by m2(i,j)= (1+(-1)^(i+j))/2, A057212. - Enrique Pérez Herrero, Jun 15 2013

Also the number of longest paths in the (n+1)-web graph for n > 2. - Eric W. Weisstein, Mar 27 2018

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Enrique Pérez Herrero, Table of n, a(n) for n = 1..1000

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), pp. 1917-1926.

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)

Eric Weisstein's World of Mathematics, Longest Path Problem

Eric Weisstein's World of Mathematics, Web Graph

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Formula

a(n) = 2*(A006918(n) + A006918(n-1) + A006918(n-2)), n>1. - Ralf Stephan, Apr 26 2003

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6), with a(1)=2, a(2)=6, a(3)=16, a(4)=30, a(5)=54, a(6)=84. - Harvey P. Dale, Apr 08 2013

From Ayoub Saber Rguez, Nov 20 2021: (Start)

a(n) = A143785(n) - A002620(n+1).

a(n) = A128624(n) + A002620(n+1).

a(n) = (n^3 + 3*n^2 + 2*n + 1 + n*(n mod 2) - ((n+1) mod 2))/4. (End)

Mathematica

Table[(1/4)*(1 + n)*(-2 + 5*n + n^2 + 2*Ceiling[1/2 - n/2] - 4*Floor[n/2]), {n, 1, 200}] (* Enrique Pérez Herrero, Aug 03 2012 *)
CoefficientList[Series[2 (1 - x^3)/((1 - x)^5 (1 + x)^2), {x, 0, 40}], x] (* Harvey P. Dale, Apr 08 2013 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {2, 6, 16, 30, 54, 84}, 40] (* Harvey P. Dale, Apr 08 2013 *)
Table[(n + 1) (2 n (n + 2) + 1 - (-1)^n)/8, {n, 20}] (* Eric W. Weisstein, Mar 27 2018 *)

Crossrefs

Essentially twice A034828.

Sequence in context: A230853 A364666 A160620 * A192735 A330503 A171218

Adjacent sequences: A005993 A005994 A005995 * A005997 A005998 A005999

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Edited by N. J. A. Sloane, Aug 03 2012

Status

approved